COEFFICIENT INEQUALITY FOR CERTAIN CLASSES OF ANALYTIC...
Transcript of COEFFICIENT INEQUALITY FOR CERTAIN CLASSES OF ANALYTIC...
Miskolc Mathematical Notes HU e-ISSN 1787-2413Vol. 17 (2016), No. 1, pp. 29–34 DOI: 10.18514/MMN.2016.768
COEFFICIENT INEQUALITY FOR CERTAIN CLASSES OFANALYTIC FUNCTIONS
T. AL-HAWARY, B. A. FRASIN, AND M. DARUS
Received 26 January, 2014
Abstract. Let T .s;�;�;ˇ/ be the class of normalized functions, f defined in the open unit diskU by
Re
��;sC1� f .´/
��;s� g.´/
!> 0 .� 2N; �;s 2N0; ´ 2U/
for some g 2Rˇ .s;�;�;ˇ/. The authors in [15] introduced the operator ��;s� defined by
��;s� f .´/D ´C
1XkD2
.kC��1/Š.��1/Š
�Š.kC��2/Šksak´
k ;
where Rˇ .s;�;�;ˇ/ denotes the class of normalized functions g in the open unit disk U definedby ˇ
ˇarg
��;sC1� g.´/
��;s� g.´/
!ˇˇ< �
2ˇ .0 < ˇ � 1/ :
For f 2 T .s;�;�;ˇ/ and given by f .´/ D ´C a2´2C a3´3C �� � ; a sharp upper bound isobtained for
ˇa3� �a
22
ˇwhen � � 1:
2010 Mathematics Subject Classification: 30C45
Keywords: univalent and analytic functions, starlike and convex functions, Fekete-Szego prob-lem
1. INTRODUCTION AND DEFINITIONS
Let A denote the family of functions of the form:
f .´/D ´C
1XkD2
ak´k; (1.1)
which are analytic in the open unit disk U D f´ W j´j < 1g: Further, let S denote theclass of functions which are univalent in U: A function f .´/ belonging to A is said
The work was fully supported by AP-2013-009.
c 2016 Miskolc University Press
30 T. AL-HAWARY, B. A. FRASIN, AND M. DARUS
to be strongly starlike of order ˇ in U , and denoted by SS�.ˇ/ if it satisfiesˇarg�´f 0.´/
f .´/
�ˇ<�
2ˇ .0 < ˇ � 1; ´ 2U/: (1.2)
If f .´/ 2A satisfiesˇarg�1C
´f 00.´/
f 0.´/
�ˇ<�
2ˇ .0 < ˇ � 1; ´ 2U/; (1.3)
then we say that f .´/ is strongly convex of order ˇ in U , and we denote by SC.ˇ/
the class of all such functions.With the help of the differential operator��;s� ;we say that a function f .´/ belongingto A is said to be in the class R
ˇ.s;�;�;ˇ/ if it satisfiesˇ
ˇarg
��;sC1� f .´/
��;s� f .´/
!ˇˇ< �
2ˇ .� 2N;�;s 2N0 DN[f0g/; (1.4)
for some ˇ.0 < ˇ � 1/ and for all ´ 2 U: Note that Rˇ.0;0;1;ˇ/ D SS�.ˇ/ and
Rˇ.1;0;1;ˇ/D SC.ˇ/.
For the class S of analytic univalent functions, [6] obtained the maximum value ofˇa3� �a
22
ˇwhen � is real. For various functions of S ; the upper bound for
ˇa3� �a
22
ˇis investigated by many different authors (see [1–5, 7, 8, 11–14, 16, 17] and [18]).
In this paper we obtain a sharp upper bounds forˇa3� �a
22
ˇwhen f belongs to
the class of functions defined as follows:
Definition 1. Let ˇ.0 < ˇ � 1/ and letf 2A. Then f 2 T .s;�;�;ˇ/ if and onlyif there exist g 2R
ˇ.s;�;�;ˇ/ such that
Re
��;sC1� f .´/
��;s� g.´/
!> 0 .� 2N;�;s 2N0 DN[f0g; ´ 2U/, (1.5)
where g.´/D ´Cb2´2Cb3´3C�� � :
Note that T .0;0;1;ˇ/DK.ˇ/ the class of close-to-convex functions defined by[3], T .0;0;1;1/DK.1/ is the class of normalized close-to-convex functions definedby [10].
2. MAIN RESULTS
In order to derive our main results, we have to recall here the following lemma.
Lemma 1 ([17]). Let h 2 P i.e. h be analytic in U and be given by h.´/ D1C c1´C c2´
2C c3´3C�� � , and Reh.´/ > 0 for ´ 2 U: Thenˇ
c2�c212
ˇ� 2�
ˇc21ˇ
2: (2.1)
COEFFICIENT INEQUALITY FOR CERTAIN CLASSES... 31
Theorem 1. Let f .´/ 2 T .s;�;�;ˇ/ and be given by (1.1). Then for ˇ � 1 and� � 1 we have the sharp inequalityˇ
a3� �a22
ˇ�ˇ2�.3s��2.�C2/�22s�.�C1/.�C1/
�22s3s.�C2/.�C1/2
C
�.3sC1��2.�C2/�22sC1�.�C1/.�C1/
�.2ˇC1/
22s3sC1.�C2/.�C1/2: (2.2)
Proof. Let f .´/ 2 T .s;�;�;ˇ/: It follows from (1.5) that
��;sC1� f .´/D��;s� g.´/q.´/; (2.3)
for ´ 2U, with q 2P given by q.´/D 1Cq1´Cq2´2Cq3´3C�� � : Equating coef-ficients, we obtain
.�C1/
�2sC1a2 D q1C
.�C1/
�2sb2; (2.4)
and.�C2/.�C1/
�.�C1/3sC1a3 D q2C
.�C1/
�2sb2q1C
.�C2/.�C1/
�.�C1/3sb3: (2.5)
Also, it follows from (1.4) that
��;sC1� g.´/D��;s� g.´/.p.´//ˇ
:
where for ´ 2 U; p 2 P and p.´/ D 1Cp1´Cp2´2Cp3´3C�� � . Thus equatingcoefficients, we obtain
.�C1/
�2sb2 D ˇp1; (2.6)
.�C2/.�C1/
�.�C1/.2/3sb3 D ˇ.p2C
3ˇ�1
2p21/: (2.7)
From (2.4), (2.5), (2.6) and (2.7) we have
a3� �a22 D
�.�C1/
3sC1.�C2/.�C1/.q2�
1
2q21/
C22sC1�.�C1/.�C1/�3sC1��2.�C2/
3sC1�22sC2.�C2/.�C1/2q21
Cˇ�.�C1/
2�3sC1.�C2/.�C1/.p2�
1
2p21/
CŒ22sC1�.�C1/.�C1/�3sC1��2.�C2/�ˇ
3sC1�22sC1.�C2/.�C1/2p1q1
C3�22sˇ2�.�C1/.�C1/�3sC1�ˇ2�2.�C2/
3sC122sC2.�C2/.�C1/2p21 : (2.8)
32 T. AL-HAWARY, B. A. FRASIN, AND M. DARUS
Assume that a3� �a22 is positive. Thus we now estimate Re. a3� �a22/, so from(2.8) and by using the Lemma 1 and letting p1 D 2rei� ; q1 D 2Rei� ;0 � r � 1;0�R � 1;0� � < 2� and 0� � < 2�; we obtain
3sC1Re.a3� �a22/
D�.�C1/
.�C2/.�C1/Re.q2�
1
2q21/C
Œ22sC1�.�C1/.�C1/�3sC1��2.�C2/�
22sC2.�C2/.�C1/2Req21
Cˇ�.�C1/
2.�C2/.�C1/Re.p2�
1
2p21/
CˇŒ22sC1�.�C1/.�C1/�3sC1��2.�C2/�
22sC1.�C2/.�C1/2Rep1q1
C3�22sˇ2�.�C1/.�C1/�3sC1�ˇ2�2.�C2/
22sC2.�C2/.�C1/2Rep21 (2.9)
�2�.�C1/
.�C2/.�C1/.1�R2/C
Œ22sC1�.�C1/.�C1/�3sC1��2.�C2/�
22s.�C2/.�C1/2R2 cos2�
Cˇ�.�C1/
.�C2/.�C1/.1� r2/
C2ˇ�22sC1�.�C1/.�C1/�3sC1��2.�C2/
�22s.�C2/.�C1/2
rRcos.�C�/
C3ˇ2Œ22s�.�C1/.�C1/�3s��2.�C2/�
22s.�C2/.�C1/2r2 cos2�
�3sC1��2.�C2/�22sC2�.�C1/.�C1/
22s.�C2/.�C1/2R2
C2ˇ�Œ3sC1��.�C2/�2sC1.�C1/.�C1/�
22s.�C2/.�C1/2rR
Cˇ
3ˇ�3s��2.�C2/�22s.�C1/.�C1/
�22s.�C2/.�C1/2
��.�C1/
.�C2/.�C1/
!r2
C�.�C1/
.�C2/.�C1/.ˇC2/
D .r;R/: (2.10)
Letting ˇ;�;� and � fixed and differentiating .r;R/ partially when �� 1;�� 0;ˇ � 1 and � � 1 we observe that
rrRR� .rR/2
D22sC4ˇ�2.�C1/2
.�C2/Œ2ˇC1��
4�3sC1�ˇ�3.�C1/
.�C1/Œ5ˇ�1� < 0:
COEFFICIENT INEQUALITY FOR CERTAIN CLASSES... 33
Therefore, the maximum of .r;R/ occurs on the boundaries. Thus the desiredinequality follows by observing that
.r;R/� .1;1/D3ˇ2
�3s��2.�C2/�22s�.�C1/.�C1/
�22s.�C2/.�C1/2
C
�.3sC1��2.�C2/�22sC1�.�C1/.�C1/
�.2ˇC1/
22s.�C2/.�C1/2:
The equality for (2.2) is attained when p1 D q1 D 2i and p2 D q2 D�2: �
Letting s D �D 0 and �D 1 in the above Theorem, we have the result given byJahangiri [9]:
Corollary 1. Let f .´/ 2K.ˇ/ and be given by (1.1). Then for ˇ � 1 and � � 1we have the sharp inequalityˇ
a3� �a22
ˇ� ˇ2.� �1/C
.2ˇC1/.3� �2/
3: (2.11)
Letting s D �D 1 and �D 0 in the above Theorem, we have the following result:
Corollary 2. Let f .´/ 2 T .1;0;1;ˇ/ and be given by (1.1). Then for ˇ � 1 and� � 1 we have the sharp inequalityˇ
a3� �a22
ˇ�1
36
�3ˇ2.3� �4/C .9� �8/.2ˇC1/
�:
ACKNOWLEDGEMENT
The authors would like to thank the referee for the informative comments andsuggestions in order to improve the manuscript.
REFERENCES
[1] H. Abdel-Gawad and D. Thomas, “A subclass of close-to-convex functions,” Publ. de L’Inst. Math.Nouvelle serie tome, vol. 16, pp. 279–290, 1985.
[2] H. Abdel-Gawad and D. Thomas, “The Fekete-Szego problem for strongly close-to-convex func-tions,” Proc. Amer. Math. Soc., vol. 114, no. 2, pp. 345–349, 1992, doi: 10.2307/2159653.
[3] A. Chonweerayoot, D. Thomas, and W. Upakarnitikaset, “On the coefficients of close-to-convexfunctions,” Math. Japon., vol. 36, no. 5, pp. 819–826, 1991.
[4] M. Darus and D. Thomas, “On the Fekete-Szego problem for close-to- convex functions,” Math.Japon., vol. 44, no. 3, pp. 507–511, 1996.
[5] M. Darus and D. Thomas, “On the Fekete-Szego problem for close-to- convex functions,” Math.Japon., vol. 47, no. 1, pp. 125–132, 1998.
[6] M. Fekete and G. Szego, “Eine Bemerkung uber ungrade schlicht Funktionen,” J. London Math.Soc., vol. 8, pp. 85–89, 1933.
[7] B. Frasin and M. Darus, “On the Fekete-Szego problem,” Inter. J. Math. and Math. Sci. (IJMMS),vol. 24, no. 9, pp. 577– 581, 2000, doi: 10.1155/S0161171200005111.
[8] R. Goel and B. Mehrok, “A coefficient inequality for certain classes of analytic functions,”Tamkang J. Math., vol. 22, no. 2, pp. 153–163, 1991.
34 T. AL-HAWARY, B. A. FRASIN, AND M. DARUS
[9] M. Jahangiri, “A coefficient inequality for a class of close-to-convex functions,” Math. Japon.,vol. 41, no. 3, pp. 557–559, 1995.
[10] W. Kaplan, “Close-to-convex schlicht functions,” Mich. Math. J., vol. 1, pp. 169–185, 1952.[11] F. Keogh and E. Merkes, “A coefficient inequality for certain classes of analytic functions,” Proc.
Amer. Math. Soc., vol. 20, pp. 8–12, 1969, doi: 10.2307/2035949.[12] W. Koepf, “On the Fekete-Szego for close-to-convex functions II,” Arch. Math., vol. 49, pp. 420–
433, 1987, doi: 10.1007/BF01194100.[13] W. Koepf, “On the Fekete-Szego problem for close-to-convex functions,” Proc. Amer. Math. Soc.,
vol. 101, pp. 89–95, 1987.[14] R. London, “Fekete-Szego inequalities for close-to-convex functions,” Proc. Amer. Math. Soc.,
vol. 117, pp. 947–950, 1993, doi: 10.1090/S0002-9939-1993-1150652-2.[15] A. Mohammed and M. Darus, “An operator defined by convolution involving the generalized
Hurwitz-Lerch zeta function,” Sains Malaysiana, vol. 41, no. 12, pp. 1657–1661, 2012.[16] M. Nasr and H. Abdel-Gawad, “On the Fekete-Szego problem for close-to-convex functions of
order p,” New Trends in Geometric Functions Theory and Applications (Madaras 1990), WorldSci. Pupl. Co., River Edge, N.J., pp. 66–74, 1991.
[17] C. Pommerenke, Univalent functions, 1st ed. Gottingen: Vandenhoeck and Ruprecht, 1975.[18] S. Trimple, “A coefficient inequality for convex univalent functions,” Proc. Amer. Math. Soc.,
vol. 48, no. 1, pp. 266–267, 1975.
Authors’ addresses
T. Al-HawaryIrbid National University, Faculty of Science and Technology, Department of Mathematics, P.O.
Box: 2600, Irbid, JordanE-mail address: tariq [email protected]
B. A. FrasinFaculty of Science, Department of Mathematics, Al al-Bayt University, P.O. Box: 130095, Mafraq,
JordanE-mail address: [email protected]
M. DarusSchool of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malay-
sia, Bangi 43600, Selangor D. Ehsan, MalaysiaE-mail address: [email protected]